MATHEMATICIANS: YESTERDAY AND TODAY*
C. T. Chong
University of Singapore
Besides teaching, part of the job at the university is to do research. In the case of
mathematics, it is mathematical research. Mathematics has had a history of more than
2500 years, and it is interesting to note how the subject develops through the course of
history.
Mathematics, indeed all sciences, was considered part of the general discipline known
as natural philosophy. In Greek times, mathematicians were sometimes also philosophers,
physicists, astronomers. Archimedes (287 - 212 B.C.), considered one of the greatest
intellect of antiquity, was known to be a physicist of the first rank. In fact until recent
times, very few leading mathematicians stayed exclusively within the bondary of mathematics as we know it today. We have among the illustrious examples people like Isaac
Newton (1642 - 1727), one of the inventors of calculus and discoverer of the law of gravitation; and Carl Friedrich Gauss (1777- 1855), great contributor to various branches of
mathematics (number theory, complex variable theory, differential geometry, statistics
etc.) and a first class astronomer.
On the other hand, it was not until the last two hundred years or so that the rnajority of creative mathematicians were associated with the universities. In those days there
were a few academies (for example the Berlin and the St. Petersberg Academies) which
were supported by royalty. Members of the academies were paid to produce scientific work.
*Based on the talks given to Hwa Chong Junior College on October 1, 1978, and to AngloChinese Junior College, on October 3, 1978, organized by the Singapore Mathematical
Society.
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But then not every mathematician was associated with such institutions. We have in mind
the case of Pierre Fermat (1601- 1665). A jurist and counsellor to the King by profession,
Fermat dabbled in mathematic is as a hobby, but his enormous contributions to the subject
place him eminently among the leaders of his time. He was, in addition to Newton and
Leibnitz (1646- 1716), a third inventor of the calculus. The many works he did in number
theory virtually steered the direction in which later research in this field was to follow.
To most people,
Fermat is best known for his 'Last Theorem', which states that
xn + yn = zn, for n > 2, has no integral solutions in x, y, z. It is now an often told story
that Fermat had scribbled in one of his books that he had found 'a truly marvelous proof
of the theorem', but the margin in the book was too small for him to include his proof.
After nearly three hundred years of concerted efforts - by professional and amateur mathematicians, and after several offers of attractive prizes for the first person to solve the
problem, the 'Last Theorem' remains today among the 'first unsolved questions' in mathematics. History has shown that Fermat's problem is a classic instance in mathematics
where pursuing a deceptively simple-looking statement could lead to a deep analysis and
significant advancement of the subject concerned. We are of course thinking about the work
of Ernst Kummer (1810 - 1893), who in his numerous attempts to solve the Fermat problem
introduced the notion of ideal numbers, marking the beginning of the theory of algebraic
numbers. It is now generally believed that Fermat's problem, if true, cannot be solved
by the elementary methods that were invented since Greek times. Even Fermat's invention
of the method of 'infinite descent' could only be used to solve some special cases of the
problem. It follows that Fermat's 'proof' of his 'Last Theorem 1 was probably false.
Things took a gradual turn for mathematicians in the late eighteenth century. By now
the majority of them were professors teaching or doing research at the universities or
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academies. Thus for example Newton spent most of his creative life in the University of
Cambridge, and Gauss was appointed Director of the Observatory at the University of
Gottingen in Germany. After his death, Dirichlet (1805- 1859) and later Riemann (18261866) succeeded the chair, and established a tradition of excellence in mathematics there
and in all of Germany.
In France all the leading mathematicians were also associated with the universities
or the French Academy - with the exception of Evariste Galois (1811- 1832), whose great
work on the theory of equations heralded the advent of group theory. Galois's case was of
course one of the tragic instances in the history of mathematics. He had completed his
revolutionary work at the age of 19, had sent copies of his manuscript to the French
Academy but received no response. And one always recalls how regrettably his disappointments led to the ending of his life (by engaging in a duel) at the young age of 21. Luckily
these things happen, when they do, only very rarely.
In any case, by the year 1850 it was common for professional mathematicians to
seek teaching jobs at the universities (sometimes even in high schools when times were
bad). These people could devote full time to their endeavours: the training of bright young
students and the pursuit of mathematical research. Their contributions to society were
recognised, and their achievements respected. In comparison with earlier periods, the
nineteenth century saw a marked increase of mathematical activities and accomplishments.
Most of the mathematics that are being taught today in the university were developed
during that time.
By the beginning of the twentieth century, the University of Gtittingen in Germany
had won the distinction of being the centre of mathematics, largely under the able leader-
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ship of Felix Klein (1849 - 1925) and David Hilbert (1862 - 1943). Aspiring and established
mathematicians from Japan, Europe and America went on pilgrimages to G"cittingen, to study
or exchange ideas with the people they had come to meet. It was around this time - at the
International Congress of Mathematicians in Paris in 1900 - that the famous 23 Hilbert
problems were presented to mathematicians of the world. These important problems covered
a wide range of subjects, and their status at each period since 1900 have often been used
as a criterion to assess the progress of mathematics. As recent as 1974 an international
symposium for this purpose was held in the U.S.A. The symposium, entitled 'Mathematical
developments arising from Hilbert problems', attempted to survey those areas of importance in contemporary mathematical research which could be traced back to the 23
Hilbert problems. To give an idea of the magnitude and significance of the contributions
of the Hilbert problems to the development of mathematics of our time, let me cite two
examples.
Hilbert's second problem dealt with, among other things, the consistency of the axioms
of number theory. Let us take the following as axioms for number theory (these are known
as the Peano axioms):
(a)
0 is a number
(b)
If n is number, then the successor of n (written n + 1) is also a number.
(c)
Different numbers have different successors.
(d)
(The Principle of Mathematical Induction). If P is a property true for O, and if
whenever it is true for n then it is true for n + 1, then P is true for all
natural numbers 0, 1, .•...•.
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Hilbert believed that the Peano axioms were consistent, i.e. one could not prove,
for example, 0
=1
from the axioms, and that the consistency could be proved from the
axioms themselves. This led to his introduction of what is now known as the Hilbert programme. The programme was directed at the solution of the consistency problem, and it
occupied him and a number of his very· able assistants !for almost two decades. The general
feeling at the 'time was that a proof of the consistency of number theory (something which
everyone believed) was only a matter of time. It therefore came as a shock in 1931 when
the 25-year-old mathematician Kurt Godel (1906 - 1978) destroyed all hope for the programmed by showing that if number theory was consistent, then the consistency could not
be proved within the system. In other words, one would have to go beyond the Peano axioms
in order to prove their consistency, if indeed they were. The implications of this result
were far reaching and profound. It is not an exaggeration to say that Godel 's proof offered
a completely new way of looking at mathematics. His work also led to the introduction of
a new mathematical field: mathematical logic, In the words of the great mathematician
John von Neumann (1903- 1957), Godel's work was 'singular in the history of logic'.
The second example is the famous Riemann Hypothesis, number eight on the list of
Hilbert problems. Riemann had, in the course of his study of the distribution of prime
numbers, introduced the zeta function
~
(s) =
where s = x + iy is a complex number. To those of you who have some acquaintance with
infinite series, you would see that ~ (s) ~ 0 for s a real number. Now the function as defined here clearly does not give a value for every s. For example, when s
= -1,
or -2
etc. the right hand side of the equation yields oo • But there is in the study of complex
function theory the notion of 'analytic continuation', wherein the function ~ (s) can be
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'cqntinued analytically' to become a 'meromorphic function' defined on the whole complex
plane. This new function coincides with the orignial one on the right-hand side of the
complex plane, i.e. on all complex numbers s
0 whenever s
=
=x
+ iy such that x > 0, and gives the value
-2, -4, •.• , - 2n, ••• Let us denote this new function again by
Riemann conjectured that except for s
the so-called 'critical line'.
= -2n, the only time when
~
(s)
=0
is when s
=!
~
(s).
+ iy,
Hilbert had called this problem 'the most important problem
in mathematics', and it remains unsolved 'today. (An analogue of the Riemann Hypothesis
in algebraic geometry has been solved by the young Belgium mathematician Pierre Deligne
(1946 -
) a few years back.)
This brings us to the work of mathematicians of our time. By and large, we are
still guided by the Hilbert problems, their ramifications and their subsequent developments.
While a number of Hilbert problems have been solved, their solutions lead to still more
pen,etrating and difficult questions. But of course not everyone works on the Hilbert problems. A rough estimate indicates that there are more than 100,000 mathematicians throughout the world, not all of them engaged in research. But the authoritative Mathematical
Reviews, published by the American ·Mathematical Society, reviews about 1500 research
papers each month, indicating that there is a great amount of mathematical research being
produced daily by mathematicians from all countries. Not all of the work done are good of
course. One would guess that perhaps many of the papers published in mathematical journals
would be forgotten soon after publication. Collectively there is nevertheless an indication
of the direction in which mathematics is going. The famous French mathematician Jean
Dieudonne (1906 mathematics'
) has recently published a paper entitled 'Current trends in pure
(AdvancesinMathematics, volume 27, 1978), giving an overview of mathe-
matics of today. About 13 mathematical disciplines were listed as belonging to the main-
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stream of mathematics, and almost all the leading mathematicians of our time work in one
or more of these areas.
While a hundred or more years ago it was not too difficult to explain to a layman
a certain mathematical problem (for example a problem in elementary number theory), it
is almost impossible today to enlighten the uninitiated on the importance of a mathematical
problem without going through a few grueling hours of definitions and motivations. This is
true especially in the case of problems in the so-called mainstream of mathematics. The
subject has advanced so fast and become ever more technical that a beginning student in
mathematics (say first year in the university) would need at least five years of intensive
study before he can embark on a research project. In some areas of mathematics it takes
an even longer time.
But there are of course always exceptional students who can finish in half the time
or less. In the history of mathematics we have examples like Gauss, Henri Poincare
(1854- 1912), von Neumann ("The fastest mind I have ever met," according to a physicist)
etc. It is on the other hand true that quick minds do not necessarily make good mathematicians, and a great mathematician need not have a quick mind. Hilbert, for example,
was known not to be particularly quick, yet his contributions certainly exceeded those of
von Neumann in depth, if not in scope.
After spending seven or eight years (or longer) learning and doing mathematics, a
student is awarded a Ph.D and is ready to pursue his own independent work. Most of the
time, however, his experience in research will be rather limited but this handicap is
compensated by the abundance of his youthful energy and enthusiasm in further improving
himself through discussions and learning. An able mathematician today can find many op-
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portunities of learning from the experts by going, if chances permit, to a mathematical
centre in Europe or America. There are establishments which have visitors from all over
the world concentrating on research, for example the Institute for Advanced Study in
Princeton, U.S.A.
To students who aspire to be mathematicians, I would suggest that hard work is the
most important pre-requisite. Gauss, certainly one of the greatest in history, firmly believed that genius was industry. A number of famous contemporary mathematicians have
expressed the view that even with only an above average intelligence, one could go quite
far by working hard. This requires dedication to the profession of mathematics. In current
economic context, there is no sense in producing a mathematician who after the Ph. D.
decides to become mathematically stagnant, unless of course he takes up a job in industry
or administration which requires a completely different kind of work. Currently there is a
surplus of Ph.D.'s in mathematics throughout the developed countries, and in some developing countries as well. It therefore seems necessary to carefully monitor the situation
in order that the bright and the dedicated be given a chance.
To the most ambitious there is the Fields Medal, awarded by the International
Mathematical Union once every four years at the opening ceremony of the International
Congress of Mathematicians. The Medal is popularly regarded as an equivalent of the Nobel
Prize in mathematics (the Nobel Foundation, of course, does not award prizes in mathematics) and is given only to mathematicians below the age of forty. But at a lower level,
a person should find great satisfaction in his work if he has done something really good.
After a period of continued frustration, the final conquest of the problem attempted usually
brings joy and pride that are difficult to describe. But of course most of the time even
a modicum of genuine success is hard to come by.
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In Singapore, whose mathematicians in the two universities have produced competent
research work during the past 30 years or so, I think that we have come a long way. But
much more work remains to be done. Perhaps one day we will be able to produce a mathematician of top calibre, something which Hong Kong has eminently achieved. This requires
time and effort, but is not an impossible task. In fact it is hoped that some of you would
take up the challenge. The going may be tough, but then nothing that is good is free, or
easy.
CORRIGENDA: "WHAT IS A STAR?" BY K.K. SEN
The following are corrections to typographical errors in the article "What is a star?"
by K.K. Sen (this Medley, Vol. 6, No. 2, August 1978, pp. 62 - 68):
p. 63, line 7 from bottom:
... Pc
0
= 28
0
GM /Rf:\••• ;
0 ~
p. 65, line 5 from bottom:
••• Conduction is too slow a process •••
p. 68, line 3 from top:
• • • transport equations (23) or
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